0.1 THE REAL NUMBER LINE AND ORDER


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1 6000_000.qd //0 :6 AM Pge 00 CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities. Use inequlities to model nd solve rellife prolems. Negtive direction ( decreses) 0 FIGURE 0..6 Positive direction ( increses) The Rel Numer Line 0 Every point on the rel numer line corresponds to one nd only one rel numer. 0 Every rel numer corresponds to one nd only one point on the rel numer line. FIGURE FIGURE 0. e π The Rel Numer Line Rel numers cn e represented with coordinte system clled the rel numer line (or is), s shown in Figure 0.. The positive direction (to the right) is denoted y n rrowhed nd indictes the direction of incresing vlues of. The rel numer corresponding to prticulr point on the rel numer line is clled the coordinte of the point. As shown in Figure 0., it is customry to lel those points whose coordintes re integers. The point on the rel numer line corresponding to zero is clled the origin. Numers to the right of the origin re positive, nd numers to the left of the origin re negtive. The term nonnegtive descries numer tht is either positive or zero. The importnce of the rel numer line is tht it provides you with conceptully perfect picture of the rel numers. Tht is, ech point on the rel numer line corresponds to one nd only one rel numer, nd ech rel numer corresponds to one nd only one point on the rel numer line. This type of reltionship is clled onetoone correspondence nd is illustrted in Figure 0.. Ech of the four points in Figure 0. corresponds to rel numer tht cn e epressed s the rtio of two integers..6 Such numers re clled rtionl. Rtionl numers hve either terminting or infinitely repeting deciml representtions. Terminting Decimls Infinitely Repeting Decimls Rel numers tht re not rtionl re clled irrtionl, nd they cnnot e represented s the rtio of two integers (or s terminting or infinitely repeting decimls). So, deciml pproimtion is used to represent n irrtionl numer. Some irrtionl numers occur so frequently in pplictions tht mthemticins hve invented specil symols to represent them. For emple, the symols,, nd e represent irrtionl numers whose deciml pproimtions re s shown. (See Figure 0..).6.96 e *The r indictes which digit or digits repet infinitely *
2 6000_000.qd //0 :6 AM Pge 0 Order nd Intervls on the Rel Numer Line One importnt property of the rel numers is tht they re ordered: 0 is less thn, is less thn., is less thn, nd so on. You cn visulize this property on the rel numer line y oserving tht is less thn if nd only if lies to the left of on the rel numer line. Symoliclly, is less thn is denoted y the inequlity <. For emple, the inequlity < follows from the fct tht lies to the left of on the rel numer line, s shown in Figure 0.. lies to the left of, so <. SECTION 0. The Rel Numer Line nd Order 00 FIGURE 0. When three rel numers,, nd re ordered such tht < nd <, we sy tht is etween nd nd write < <. is etween nd. The set of ll rel numers etween nd is clled the open intervl etween nd nd is denoted y,. An intervl of the form, does not contin the endpoints nd. Intervls tht include their endpoints re clled closed nd re denoted y,. Intervls of the form, nd, re neither open nor closed. Figure 0. shows the nine types of intervls on the rel numer line. Open intervl Intervls tht re neither open nor closed Infinite intervls (, ) (, ] (, ) (, ) < < < < > Closed intervl [, ] [, ) < (, ] [, ) (, ) FIGURE 0. Intervls on the Rel Numer Line STUDY TIP Note tht squre rcket is used to denote less thn or equl to or greter thn or equl to. Furthermore, the symols nd denote positive nd negtive infinity. These symols do not denote rel numers; they merely let you descrie unounded conditions more concisely. For instnce, the intervl, is unounded to the right ecuse it includes ll rel numers tht re greter thn or equl to.
3 6000_000.qd //0 :6 AM Pge 00 CHAPTER 0 A Preclculus Review STUDY TIP Notice the differences etween Properties nd. For emple, < < nd < >. ALGEBRA REVIEW Once you hve solved n inequlity, it is good ide to check some vlues in your solution set to see whether they stisfy the originl inequlity. You might lso check some vlues outside your solution set to verify tht they do not stisfy the inequlity. For emple, Figure 0.6 shows tht when 0 or the inequlity is stisfied, ut when the inequlity is not stisfied. Solving Inequlities In clculus, you re frequently required to solve inequlities involving vrile epressions such s <. The numer is solution of n inequlity if the inequlity is true when is sustituted for. The set of ll vlues of tht stisfy n equlity is clled the solution set of the inequlity. The following properties re useful for solving inequlities. (Similr properties re otined if < is replced y nd > is replced y. ) Properties of Inequlities Let,, c, nd d e rel numers.. Trnsitive property: < nd < c < c. Adding inequlities: < nd c < d c < d. Multiplying y (positive) constnt: < c < c, c > 0. Multiplying y (negtive) constnt: < c > c, c < 0. Adding constnt: < c < c 6. Sutrcting constnt: < c < c Note tht you reverse the inequlity when you multiply y negtive numer. For emple, if <, then >. This principle lso pplies to division y negtive numer. So, if >, then <. EXAMPLE Solving n Inequlity Find the solution set of the inequlity <. For = 0, (0) =. For =, () =. For =, () = 8. 0 Solution set for < FIGURE SOLUTION < Write originl inequlity. < Add to ech side. < 9 Multiply ech side y < 9. < So, the solution set is the intervl,, s shown in Figure 0.6. TRY IT Find the solution set of the inequlity <. In Emple, ll five inequlities listed s steps in the solution hve the sme solution set, nd they re clled equivlent inequlities.
4 6000_000.qd //0 :6 AM Pge 0 The inequlity in Emple involves firstdegree polynomil. To solve inequlities involving polynomils of higher degree, you cn use the fct tht polynomil cn chnge signs only t its rel zeros (the rel numers tht mke the polynomil zero). Between two consecutive rel zeros, polynomil must e entirely positive or entirely negtive. This mens tht when the rel zeros of polynomil re put in order, they divide the rel numer line into test intervls in which the polynomil hs no sign chnges. Tht is, if polynomil hs the fctored form r r < r < r <... r,..., r n, < r n then the test intervls re, r, r, r,..., r n, r n, nd r n,. For emple, the polynomil 6 cn chnge signs only t nd. To determine the sign of the polynomil in the intervls,,,, nd,, you need to test only one vlue from ech intervl. SECTION 0. The Rel Numer Line nd Order 0 EXAMPLE Solving Polynomil Inequlity Find the solution set of the inequlity < 6. SOLUTION 6 < 0 < 0 Write originl inequlity. Polynomil form Fctor. So, the polynomil 6 hs nd s its zeros. You cn solve the inequlity y testing the sign of the polynomil in ech of the following intervls. <, To test n intervl, choose representtive numer in the intervl nd compute the sign of ech fctor. For emple, for ny <, oth of the fctors nd re negtive. Consequently, the product (of two negtive numers) is positive, nd the inequlity is not stisfied in the intervl <. A convenient testing formt is shown in Figure 0.. Becuse the inequlity is stisfied only y the center test intervl, you cn conclude tht the solution set is given y the intervl < <. < 6 < <, > Solution set Sign of Sign 0 < 0? No No Yes 0 Yes Yes Yes 0 No No No Yes No ( )( ) > 0 ( )(+) < 0 (+)(+) > 0 FIGURE 0. Is < 0? TRY IT Find the solution set of the inequlity > 0.
5 6000_000.qd //0 :6 AM Pge CHAPTER 0 A Preclculus Review Appliction Inequlities re frequently used to descrie conditions tht occur in usiness nd science. For instnce, the inequlity W 80 descries the recommended weight W for mn whose height is feet 0 inches. Emple shows how n inequlity cn e used to descrie the production level of mnufcturing plnt. EXAMPLE Production Levels In ddition to fied overhed costs of $00 per dy, the cost of producing units of n item is $.0 per unit. During the month of August, the totl cost of production vried from high of $ to low of $00 per dy. Find the high nd low production levels during the month. SOLUTION Becuse it costs $.0 to produce one unit, it costs. to produce units. Furthermore, ecuse the fied cost per dy is $00, the totl dily cost of producing units is C. 00. Now, ecuse the cost rnged from $00 to $, you cn write the following Write originl inequlity. Sutrct 00 from ech side. Divide ech side y So, the dily production levels during the month of August vried from low of 80 units to high of 0 units, s shown in Figure Ech dy s production during the month fell in this intervl. Low dily production FIGURE High dily production TRY IT Use the informtion in Emple to find the high nd low production levels if, during Octoer, the totl cost of production vried from high of $00 to low of $000 per dy.
6 6000_000.qd //0 :6 AM Pge 0 SECTION 0. The Rel Numer Line nd Order 0 EXERCISES 0. In Eercises 0, determine whether the rel numer is rtionl or irrtionl. * e In Eercises, determine whether ech given vlue of stisfies the inequlity.. > 0 () () (c) (d). < () 0 () (c) (d). 0 < < () () 0 (c) 0 (d). < () 0 () (c) (d) In Eercises 8, solve the inequlity nd sketch the grph of the solution on the rel numer line.. 6. >. < 8. < 9. < 0.. < <. 0 <.. < > > <. 6. > >. < 6 8. < 9 *The nswers to the oddnumered nd selected even eercises re given in the ck of the tet. Workedout solutions to the oddnumered eercises re given in the Student Solutions Guide. 9. Biology: ph Vlues The ph scle mesures the concentrtion of hydrogen ions in solution. Strong cids produce low ph vlues, while strong ses produce high ph vlues. Represent the following pproimte ph vlues on rel numer line: hydrochloric cid, 0.0; lemon juice,.0; oven clener,.0; king sod, 9.0; pure wter,.0; lck coffee,.0. (Source: Adpted from Levine/Miller, Biology: Discovering Life, Second Edition) 0. Physiology The mimum hert rte of person in norml helth is relted to the person s ge y the eqution r 0 A where r is the mimum hert rte in ets per minute nd A is the person s ge in yers. Some physiologists recommend tht during physicl ctivity person should strive to increse his or her hert rte to t lest 60% of the mimum hert rte for sedentry people nd t most 90% of the mimum hert rte for highly fit people. Epress s n intervl the rnge of the trget hert rte for 0yerold.. Profit The revenue for selling units of product is R.9 nd the cost of producing units is C 9 0. To otin profit, the revenue must e greter thn the cost. For wht vlues of will this product return profit?. Sles A doughnut shop t shopping mll sells dozen doughnuts for $.0. Beyond the fied cost (for rent, utilities, nd insurnce) of $0 per dy, it costs $. for enough mterils (flour, sugr, etc.) nd lor to produce ech dozen doughnuts. If the dily profit vries etween $0 nd $0, etween wht levels (in dozens) do the dily sles vry?. Reimursement A phrmceuticl compny reimurses their sles representtives $0. per mile driven nd $00 for mels per week. The compny lloctes from $00 to $0 per sles representtive ech week. Wht re the minimum nd mimum numers of miles the compny epects ech representtive to drive ech week?. Are A squre region is to hve n re of t lest 00 squre meters. Wht must the length of the sides of the region e? In Eercises nd 6, determine whether ech sttement is true or flse, given <.. () < 6. () < () < () < (c) 6 < 6 (c) < (d) < (d) <
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